RESIDUAL ERRORS IN ALTIMETRY DATA
DETECTED BY COMBINATIONS OF
SINGLE- AND DUAL-SATELLITE CROSSOVERS

version "comb_996.tex ", 29th July '99

Jaroslav Klokočník (2), Carl A. Wagner (1), Jan Kostelecký (3),

(1) NOAA, 1305 East-West Highway, Silver Spring, MD-20910, USA.
(2) Astronom. Inst. Czech Acad. Sci., CZ-251 65 Ondřejov Observatory
(3) Faculty of Civil Eng., CTU Prague, CZ-160 00 Prague 6, Thákurova 7 and Research Institute of Geodesy Topography and Cartography, CZ - 250 66 Zdiby.

ABSTRACT

The Single- and Dual-Satellite Crossover (SSC and DSC) residuals between and among Geosat, T/P, and ERS 1 or 2 have been used for various purposes, applied in geodesy for gravity field accuracy assessments and determination as well as in oceanography. Here we present the theory and give various examples of certain combinations of SSC and DSC that test for residual altimetry data errors, mostly of non-gravitational origin, of the order of a few centimeters. There are 4 types of the basic DSCs and 12 independent combinations of them in pairs which we have found useful in our work. We define them in terms of the "mean" and "variable" components of a satellite's geopotential orbit-error from Rosborough's 1st order analytical theory. The remaining small errors, after all altimeter data corrections are applied and the relative offset of coordinate frames between altimetry missions removed, are statistically evaluated by means of the Student distribution. The remaining signal of 'non-gravitational' origin can in some cases be attributed to the main ocean currents which were not accounted for among the media or sea surface corrections. In future, they may be resolved by a long-term Global Circulation Model. We describe our experience with two current one's (Stammer et al., 1996 and Carton, 1996) neither of which are found either to cover our most critical missions (Geosat & TOPEX/Poseidon) or to have the accuracy and resolution necessary to account for the strongest anomalies found across them. In other cases, the residual signal is due to errors in tides, altimeter delay corrections or El Niño. (We present various examples of these as well.) Our tests of the combinations of the JGM 3-based DSC residuals show that overall the long-term data now available are well suited for a gravity field inversion refining JGM 3 for low and resonant order geopotential harmonics whose signatures are clearly seen in the basic DSC and SSC sets.

Key words. Satellite altimetry, Single- and Dual- crossovers, Student-t statistics.

1. THEORETICAL PART

1.1. Introduction

"If the change of global mean sea level is to be accurately measured by satellite altimeter, the consistency and accuracy of all aspects of the measurement process, including the complex of media corrections and orbit models, will have to be assured" (Wagner and Cheney, 1992). There are not only "traditional" media corrections, sea state bias, inverse barometer correction, the 1 cpr empirical orbit correction, etc., but also a new complication, unexpected before: there appears to be relative coordinate system offset in our cross-mission (DSC) altimetry data, though how this occurs is not fully understood (Wagner et al, 1997b).

In our previous papers (Wagner et al, 1997a,b), we have developed and numericaly tested algorithms for the identification and removal of these offsets and applied them to the most egregious case, namely the DSCs between the less accurate Geosat and the more accurate TOPEX/Poseidon (T/P) missions. To interconnect multiple altimetry missions (and assess as long an oceanographic series of data as possible), this correction must be removed before oceanographic application can take place. Numerically, extraction of the offset may pose a problem however, as the forbidden harmonics (representing the relative offset) inevitably correlate with the higher degree first order geopotential harmonic coefficients (e.g., Bosch et al, 1998) that are also uncertain in the orbit model used to generate the altimeter heights.

In this paper, we investigate residual altimetry errors by using certain combinations of the dual-satellite crossovers (DSC), offset-free when necessary. We make use of long-term averaged (over 1-3 years) crossover data. We assume that the reader is familiar with the definitions of the single-satellite crossovers SSCs and the DSCs (Shum et al, 1990; Klokocnik et al, 1993) and with the Rosborough-Engelis analytical theory (Rosborough 1986, Engelis 1987, Rosborough and Tapley, 1987).

1.2. Twelve Combinations of the DSCs

Definitions

Rosborough (1986) showed explicitly that and how the radial orbit component is decomposed into two parts, , geographically correlated part (does not depend on the sense of the orbit track), sometimes called the 'mean', and , called the 'variable' or anticorrelated part (of the same magnitude but opposite sign for the ascending and descending orbit segments). The Single Satellite Crossovers (SSCs) in terms of Rosborough's theory (Rosborough, 1986) are:

where A, D denote the Ascending and Descending tracks, respectively, and is the variable part. The Dual-Satellite Crossovers DSCs (AD, DA, AA and DD) are (Klokocnik et al, 1993):

where the lower indices "1"/ "2" belong to the first/second orbit of the satellite's pair (e.g. ). Usually (but not necessarily), the first satellite in the pair is 'low' flying, the second is 'higher' flying (e.g., Geosat & Topex).

In the following text, we use more concise notation, e.g. instead , we write "AD"; the first/second capital for the respective track of the first/second satellite of the given pair. There are just 12 independent combinations of the DSCs, taken in pairs (Klokocnik and Wagner, 1999):

The combinations with plus sign are close substitutes for either a full radial component or the mean (correlated) radial part of the lower orbit; we call these combinations substituting. The combinations with minus sign are candidates for zero gravity effect if a remainder from the right hand sides is subtracted; this can be done by means of independently derived SSCs (see eq. 17 below). We call these, potentially zero combinations (Klokocnik and Wagner, 1999). (When the subtraction is not done we call them DSC surrogates for the SSCs and we illustrate their use in detecting non-gravitational effects in altimetry below). A sub-category of these potential zeros, with - which is small if the second satellite is 'high' flying - has very small gravity effect even without any subtraction from SSCs (eqs. 4 and 6); we call these combinations 'levitating', or lightheaded, being residual smaller perturbations of the upper orbit, levitated by the cancelled stronger effects on the lower satellite.

Geocenter Offset-Free Combinations

It is known that the DSCs can interconnect (two or more) satellite altimetry missions into one 'geodetic system' (e.g., Wagner and Cheney, 1992), even over a decade, provided we know their reference-frame constants precisely. The DSCs have been used (Naeije et al, 1996; Wagner et al, 1997a,b) to detect relative coordinate system offsets in the missions Geosat vs T/P or ERS 1 vs T/P. Thus, the DSCs - in contrast to the SSCs - are sensitive to the 'forbidden' harmonic coefficients , , (Wagner et al 1997b). We use a generalization of (2), as a single equation:

where - ,
i is A or D,
j is D or A.
.

If i=j (i.e. for "AA" or "DD"), then = - 1,

if (i.e. for "AD" or "DA"), then = + 1.

If i=D (the first upper index is "D"), then ,
if i=A, then .

Each DSC residuum consists of the gravity-orbit part (2) and from the relative offset , meaning or . Thus, eqs. (2) or (15) can be rewritten as

where T means the total value of the DSC residuum. The combinations (3)-(14) contain if there is a sum of DSCs, but they are offset-free if there is a difference of the DSCs.

Check of SSCs by DSC combinations.

It follows from (3)-(14), by using (1) that

,

,

,

,

,

where the are independently computed single crossover residuals. [Similar relationships can also be generated for the sums of AA, DD, DA, or AD, but they suffer from the offset].

We see that the combinations of the DSCs can be verified independently by means of or or by their sum or difference. Remarkable are those equations with , as they should lead to numerically small values. Thus, we have a diagnostic tool to investigate remaining systematics in the altimetry data, e.g., (DD - DA) - = 0. We also see immediately that (AA - DA) + (DD-AD) = 0. In practise, the combinations will not be exactly zero and the non-zero value will be an error diagnostic. [But note that the total 'noise' in these combinations summed over the contributions from the individual AA, DA, DD, and AD residuals may become so large that a unique interpretation of systematic error may not be possible.]

2. STATISTICAL TREATMENT

We found interesting types of the combinations of the DSCs, namely the substituting and zero combinations (Klokocnˇk and Wagner, 1999). The first type can be applied to dual altimetry missions in high-low orbits (like Geosat and T/P), and yields for example approximate representation of the dominant 'mean' part of the full radial perturbations of the lower orbit. The second type yields no orbit error due to the static geopotential and can help to elucidate oceanographic changes between passes, among other media errors in the altimetry corrections.

We call eqs. (3) - (14) basic diagnostics equations. To use them for numerical computations and statistical assessment of the crossover residuals, they need to be slightly modified.

Let us consider for example eq. (13) with AA and DD already "offset free" (meaning that the relative coordinate system offset has already been removed from AA and from DD):

where 'a gravitational signal' comes from the static geopotential (represented by a gravity field model). In reality, eq. (18) will not be zero and the value

can be estimated statistically to see if the residual is likely to contain a 'non-gravitational bias' (i.e. a signal from other than the static geopotential; for example, tidal error). To turn eq. (18) into a "diagnostic" for the 'non-gravitational bias' we assume and . We estimate these expectations by projecting the calibrated variance-covariance matrix of the gravity model used for the orbit into the indicated crossover quantities, yielding their standard errors of the relevant components (for the formulae see Kloko‡nˇk et al, 1993). In our case, we make use of the JGM 3 covariances and the software developed earlier for their projections to the crossover errors (Kloko‡nˇk et al, ibid).

But what do we know about the validity of the JGM 3 covariance matrix? Besides the extensive calibrations of it by its authors on ordinary satellite data (Tapley et al, 1996) we have made our own assessment (in terms of altimetric lumped latitude coefficients) and conclude that for Geosat, ERS 1 and T/P, the estimated errors of JGM 3 from its covariance matrix are reasonable for the most part (Kloko‡nˇk et al, 1998, 1999). Indeed our tests here are additional confirmations of this calibration in terms of the geographic representation of the DSCs and SSCs.

So, for the purpose of the statistical treatment of (19) we use - point by point (crossover by crossover) - the residual value corresponding to in this way

and its error estimate

where are the formal standard deviations of AA and DD (assuming no correlation between them) with degrees of freedom of AA and DD, respectively, and are the standard errors projected from the JGM 3 covariances of the harmonic coeffcients (for the formulae see Klokocnik et al, 1993).

For the diagnostics given by eqs. (17), which are "potential zero's" not requiring a geopotential covariance projection from a gravity model, we have, by analogy to eqs (20) and (21), for example

and the error estimate

with , , and degrees of freedom of the DD, DA, and input residuals.

The ratio

has the character of the Student-t distribution. For plotting figures we can use the ratio

where the values of the Student distribution for risk =1% are taken from a table (for this distribution) and the degree of freedom from our data files of the DSC residuals. The ratio means that we accept the null hypothesis (i.e. that the measured is random with variance ), while means that we reject the zero (random) hypothesis and suggest the measured is anomalous. For higher and higher n, the Student distribution is closer and closer to the normal distribution (e.g., for n = 5, 30, 100, 300, =1%, are 4.03, 2.75, 2.63, 2.59, etc, respectively, in contrast to the normal distribution, where the interval is constant, for the given risk; here 2.50). For n lower than 10, we do not expect that the statistic is conclusive for our purpose.

3. EXAMPLES

Due to space reasons we cannot present all the examples we have available. For these the reader can contact the authors by e-mail and/or to visit pub directory on sunkl.asu.cas.cz and can GET by anonymous FTP various (mostly) color plots (the directory: /pub/jklokocn/GMTFIG/ and /pub/jklokocn/PAPERS/JG4_99/). We used GMT (Wessel and Smith, 1995), version 3.0.

To avoid any misinterpretation of the following examples, we emphasize:
(1) Case : we are nearly sure (99%) that the null hypothesis can be rejected, i.e. the DSC combinations exhibit a systematic effect (not covered by errors in the JGM 3 model and by "random" errors of the DSC data themselves).
(2) Case : a systematic effect (probably due to a 'non-gravitational' error) may exist in the tested DSC combination, but it is smaller than the error permitted in the gravity model used plus crossover data noise.

3.1. Data and JGM 3 Covariance Projections

We use NOAA GDR files for Geosat, NASA/JPL GDRs for T/P and ESA ODRs for ERS 1. The orbit basis for the altimetric heights for the Geosat and T/P files is the JGM 3 geopotential (Tapley et al, 1996). We used NOAA Geosat SSCs and DSCs from both GM and ERM mission. The basis for ERS 1 heights is also JGM 3 for an early release (1996) but the model DGM E04 (Scharroo and Visser, 1998) was used for Pathfinder data in 1997 which we also employed in our comparisons. (We converted the ERS 1 heights from DGM E04 to JGM 3 basis by applying to them, according to pass sense, the geopotential-orbit effect from the difference in the two models). The crossovers were computed at NOAA, Silver Spring, from algorithms by one of the authors (CW) for Pathfinder data, and Russ Agreen for data from the other sources.

The original altimeter heights for all three missions were first reduced to sea surface heights using the reference orbits and a standard ellipsoid and geoid for all three missions after correction for path delays and biases from a number of media sources. These corrections include delays from the ionosphere, the wet and dry troposphere, the sea state bias and a variable ocean surface model which includes the inverse barometer response to air pressure as well as the ocean tides from the CSR3.0 model (including bottom load) and the solid earth tides (see Table 2 in Kloko‡nˇk et al, 1999). An important change in the sea state bias for T/P occured between an earlier (1996) release and one employed in 1997 (Pathfinder). The significance of these discrepancies will be noted below.

Figs. 1 and 2 show the DSC residuals (in [cm]), the data input to our analysis, for Geosat&T/P and for ERS1&T/P JGM 3-based, respectively. All AD, DA, AA, and DD sets (except AD and DA for Geosat&T/P) have been cleared of a relative coordinate frame offset between Geosat or ERS 1 and T/P. (There is insufficient data in the tropics to clear AD and DA of this offset for Geosat&T/P). Data shown on Figs. 1 and 2 are long-term averages, in this case 1 year averaged month-to-month data. The 3 year averaged data for Geosat&T/P look similarly and has already been shown elsewhere (Kloko‡nˇk et al, 1998). The term 'month-to-month' average over a x-year gap between years and means we average data from January of satellite with data from January of satellite , Feb of with Feb of , etc. We hope to avoid in this way the seasonal fluctuations of both the ocean's surface and media errors as much as possible.

The projections of the calibrated variance-covariance matrix of JGM 3 - the model used to compute the orbits and all the crossovers - are shown for Geosat, T/P, and ERS 1 respectively on Figs. 3 and 4; always only those parts that we need to perform the Student statistics on the relevant diagnostic equations. The formulae are taken from (Kloko‡nˇk et al, 1993).

3.2. (AA-DD) and (AA+DD) combinations

Figs. 5a,b,c,d show AA-DD and AA+DD of Geosat&T/P from the 1 year averaged data and the Student statistics (23) on the diagnostic equations (14) and (13). While for the difference (14), Fig. 5c, there is no remaining systematic effect, for the sum (13), on Fig. 5d, we see anomalous residual features mostly along the main ocean currents. Recall that there is an 8-year gap in the Geosat&T/P DSCs and these effects seem to be unaveraged anomalies for these active currents. Note that the pair sum tends to keep any such environmental anomalies while the pair difference tends to cancel them. The reason is the difference between ascending and descending times and locations in the bins for each month is negligible compared to the gross environmental effect over the multi-year gap.

The artefact west of Australia on Fig. 5d may be a small geopotential-orbit anomaly above the 1- level of JGM 3 harmonic coefficients, or due to problematic tide modelling in these seas (Chao 1998, priv. commun.), but also, considering the time span of the Geosat data (April 1985 - March 1986), it could be an un-averaged El Niño effect in the Indian ocean. Another possible origin would be meanders in the Leeuwin current which is flowing in this area and is poorly known.

To assess the role of averaging, we add Fig. 5e with the same Student statistics for the AA+DD Geosat&T/P data, but from 3-year averages (month-to-month averages leading to minimization of seasonal effects). Both populations (the 3 y and 1 y averages) have similar statistical properties (distribution and formal standard deviations). By comparing Fig. 5e to 5d, we still see anomalous features in the north-west Pacific ocean and along other main ocean currents, but to a lesser extent than from the 1 y averages, much less anomaly west of Australia from the 3 y averages, but more pronounced differences in a tropical Pacific area (two 'hills'). Our explanation is that the longer averaging helps to decrease the residuals due to the unmodelled ocean currents; the mid-Pacific 'hills' are echos of El Niño, by chance missing in the 1 year averages (which were computed from Geosat-T/P data in the interval 1985Apr1993 to 1986Mar1994).

3.3. On the ocean currents removal

For geopotential analysis, a logical further step would be to model the ocean currents, providing a new correction of altimetry data due to them, and to remove this source from the DSC residuals. Then, with new, repeated tests with the combinations of the DSC and the relevant Student statistics, we should see a surface as in Fig. 5d or 5e, but without systematic effects.

Is long term ocean modelling currently up to this task? Our conclusion is yes (with significant uncertainties) for certain time periods, time differences and ocean areas, but no specifically for the 8 year gap in our Geosat&T/P DSCs or for the zones of strong gyres and currents at mid to high latitudes.

We had available and analysed the behaviour of two Ocean Circulation Models (OCM), POCM (Parallel Ocean Climate Model, version  4B; Stammer et al, 1996) and Carton's OCM (Carton, 1996). Both models are generated from so-called primitive equations operating at many depth levels using algorithms first developed more than 20 years ago at Princeton's Geophysical Fluid Dynamics Laboratory. The essential differences between them are that POCM is an eddy resolving model, important for our anomalous data, but without external data assimilation, while Carton's OCM does not have high resolution at mid to high latitudes but does assimilate global (collinear) satellite altimetry as well as heat data in the tropics. However, most important for us, POCM  4B is only available from 1987 through 1996 while Carton's GCM only covers 1980 through 1995 and so both are not available for our full Geosat-T/P DSCs (1985-1996).

Nevertheless the OCMs are promising and show rather good correlation with sea height differences when averaged over extended spaces and times. This is best illustrated in global comparisons with collinear T/P altimetry taken over seasonal time spans (Figures 6 a,b,c). Here we show average sea height differences of the models and T/P altimetry between the months of September and March 1993, in 2x4 degree bins for T/P (latitude by longitude), 1x1 for POCM 4B and with variable bin sizes for Carton's OCM (2.5 in longitude by 0.5 in latitude at the equator to 3 at latitude 60 degrees). While the appearance of correlation between the models and the altimetry is good, especially in the northern hemisphere, the actual performance globally is only fair. The correlation of the POCM 4B with the altimetry is actually much more significant than for Carton's OCM even though the overall value is somewhat less because many more POCM bins are covered in the comparison. In addition, the POCM 4B contains no altimeter data.

We have also tested these models against revolutions of T/P collinear altimeter differences (point by point along track) across maximum season contrasts (such as those above) and find the correlation of model and altimetry to be positive but even less so than for the global area monthly averages. Our conclusion is that except for certain (quieter) areas the models do not have the accuracy necessary for geodetic use.

3.4. Substituting and Levitating combinations

Figs. 7a,b,c,d show (for ERS 1&T/P) examples of substituting combinations AD+AA and DD+DA and Figs. 8a,b,c,d of levitating combinations AD-AA and DD-DA, with their relevant statistics. In both cases we can see 'along-track' residual patterns following the ascending or descending tracks of ERS 1, but their origin on Figs. 7 and 8 must be different. The anomalies in the sum "substitutions" can combine (add) anomalous orbit-geopotential error in ERS 1 from JGM 3 with environmental (media correction) errors from both altimeters while those in the minus "substitutions" should be restricted only to media or tidal correction errors in the T/P part of the pairs. But the geopotential-orbit contribution for T/P is minor, about 1 cm; see Figs. 3c,d. We see that Figs. 8c,d exhibit strong systematic effects - the levitating combinations, however, should be very small (see eqs. 4 and 6); in turn, Figs. 8c,d indicate possible residual errors in media or tidal corrections for T/P, in the original NOAA GDR data (see below for more details at Figs.  9 and 10).

3.5. Zero combinations, using DSC - SSC Residuals

Assume a 'high-low' altimetry satellite pair, like ERS&T/P or Geosat&T/P. It is probably expected by readers that the SSCs, DSCs and their combinations will be used to show some 'defects' or 'inaccuracy' for the lower orbiting satellite. On the contrary by focusing on the geopotentially quieter higher orbit we may more easily detect smaller non-geopotential biases. We recall eqs. (1), (4), (6) or (17). From these it follows that

But recall also that belongs in this case to T/P. Therefore, by using the levitating combinations, we can 'filter-out' the effect on the lower satellite and test the higher flying satellite. For example, we can compare DSC from Geosat&T/P or ERS&T/P to SSC of T/P, determined independently. We have several such examples for NOAA and GFZ data, revealing a remarkable difference: a systematic trend of SSC T/P from the NOAA data and only a noise in T/P SSCs from GFZ crossover reduction (for more details see Kloko‡nˇk et al, 1998). Here we present another example, Fig 9  a,b,c, where we even use DSCs based on two different gravity models!

Fig. 9 a,b show the levitating combinations (DD-DA) and (AD-AA) of ERS1 & T/P (both surrogates for T/P SSCs), where the orbit of ERS 1 is based on the DGM  E04 model, tailored to ERS1 & 2 orbits (by their SSCs), see Scharroo and Visser (1998), but the T/P passes are JGM 3-based. Fig. 9c then shows the SSCs of T/P, JGM 3-based. Note that while the SDs of the combinations must be expected to be higher than the SDs of the SSCs (1.7 cm vs 1.0 cm in this case), the values of the combinations themselves are larger than their SDs and indeed diagnose a 'non-gravitational' bias.

The agreement between Figs. 9a and 9b is very good; the same is true for Figs. 9a&b vs Fig. 9c (i.e. DSCs combinations vs SSC of T/P), although the DSCs are 'internally inconsistent' i.e. related to two different gravity field models. According to the theory, we deal with the zero combinations and thus, that 'inconsistency' is cancelled exactly. [In reality, the cancelation can never be exact because the location of the crossovers in the bins differ slightly between the various types].

We have in fact SSCs of T/P derived twice: (i) once from the levitating (surrogate) combinations, based on the DSCs (Figs.  9a,b); these are the 'surrogates' of T/P SSC, according to eqs. 4, 6, and 17; (ii) once from the actual SSC data of T/P (Fig.  9c), independent of the DSC combinations. Consider also Fig. 10, where 10a shows the NOAA (AD-AA) DSC combination, and Fig. 10b the Pathfinder (AD-AA) DSC combination. These 'surrogates' of T/P SSCs are compared to the actual SSC of T/P, now on Fig.  10c, using the SSC from the NOAA data, and on Fig.  10d, from the Pathfinder data, (all JGM 3-based).

The two direct sources of T/P SSC are obviously showing nearly the same result and this result agrees with the Pathfinder surrogate, not the NOAA. The NOAA DSCs must to be blame and since the surrogates isolate the signal from T/P part (the ERS  1 part cancelling), this proves it is the T/P of the NOAA DSCs that are the cause of the problem. Note also that the Student statistics applied to the surrogate combinations (AD-AA) or (DA-DD) with the NOAA data confirms systematics effects (see Figs. 8c,d) while with the Pathfinder data, there are no such systematic errors (not shown here).

In summary (to subsection 3.4. and 3.5), we confirm our previous suspicion (from Fig.  8) that there is a problem in the older NOAA data for T/P. Indeed, it is known that the 'media model' for T/P at NOAA was changed after 1996 (principally the electromagnetic bias) and improved (Kuhn, 1998, private commun.).

This example is not presented here to criticize NOAA for their process of correcting altimetry data, but to show the possibility of our method detecting discrepancies in residual altimetry signals even for T/P.

4. CONCLUSION

Combinations of the single and dual-satellite crossovers (SSC and DSC) in altimetry have been found useful for error analysis in conjuction with their application to geodesy and oceanography. One kind of the combinations, when applied to dual missions in high-low orbits, approximates the dominant 'mean' part or the total radial perturbations of the lower orbit. Another kind yields no geopotential error and can help elucidate oceanographic changes between passes, among other errors of altimetry corrections. A third kind is a surrogate of SSC data and can be used to check the significance of the background (non-geopotential) errors that are not common to the two kinds of crossovers. We demonstrate, with this last kind, a significant inconsistency in the media corrections recommended for T/P (NOAA data) prior to 1997 with those after.

We have found that certain crossover combinations show the residual unmodelled effect of the ocean currents over long time periods. Our attempt to employ existing Ocean Current Models for additional correction of altimetry data have not proved successful because the models do not yet cover adequately the time spans and ocean areas involved.

Acknowledments We are grateful to Prof. Francois Barlier for his care with our manuscript and two anonymous reviewers for their comments. We thank John Lillibridge and John Kuhn for the generation of NOAA's Geophysical Data Records and many useful consultations about their contents, Brian Beckley for essential and continuing help with interpreting the Pathfinder data, Laury Miller for providing access to the Carton's OCM, and Tom Johnson for kindly supplying his extensive bin averages of the POCM  4B surface heights. The support by grant A 3003703 from the Grant Agency of the Academy of Sciences of the Czech Republic (for JK &JK) is gratefully acknowledged. This research has been performed in the frame of Key project K 1003601 and project No. 7 of the Faculty of Civil Eng., Czech. Techn. Univ., Prague. We also thank Walter Smith and Paul Wessel for the GMT 3.0 version (Generic Mapping Tools) and consultations.

References

Bosch W, Klokocnik J, Wagner CA, Kostelecky J (1998) Geosat and ERS -1 Datum Offsets Relative to TOPEX/Poseidon and Geopotential Corrections Estimated Simultaneously from Dual-satellite Crossover Altimetry, presented at IAG Sect. II Symp. "Towards an Integrated Global Geodetic Observing System", Munich, Germany.

Carton, J (1996) Analysis of Global Ocean Assimilation,
www.meto.umd.edu/~carton/index.html.

Engelis T (1987) Radial Orbit Error Reduction and Sea Surface Topography Determination using Satellite Altimetry, Dept. Geodet. Science and Surveying, Ohio State Univ. Rep. No. 377, Columbus.

Klokocnik J, Kostelecky J, Jandova M (1993) Altimetry with Dual-Satellite Crossovers, presented at IAG GM Beijing, China, Session D 10: "Impact of Satellite Altimetry on Gravity Field Determination", see also (1995): Manuscr. Geod., 20, 82.

Klokocnik J, Wagner CA, Kostelecky J (1996) Accuracy Assessment of Recent Earth Gravity Models using Crossover Altimetry, Studia Geophys. et Geod., 40, 77.

Klokocnik J, Wagner CA, Kostelecky J, Rentsch M (1998) Residual Errors in Dual-Satellite Crossover Altimetry Data: An Independent Check, presented at EGS GA, Symp.: Ocean Modelling from Altimetry and Remote Sensing, Nice, France.

Klokocnik J, Wagner CA, Kostelecky J (1999) Spectral Accuracy of JGM 3 from Satellite Crossover Altimetry, J. Geod., 73, 138.

Klokocnik J, Wagner CA (1999) Combinations of Satellite Crossovers to Study Orbit and Residual errors in Altimetry, Celest. Mechanics, in print.

Naeije MC, Scharroo R, Wakker KF (1996) Monitoring of Ocean Current Variations, NUSP Rep. 96-08, Delft Univ. Techn., Delft Inst. for Earth Oriented Space Research.

Rosborough GW (1986) Satellite Orbit Perturbations due to the Geopotential, CSR-86-1 Univ. of Texas at Austin, Center for Space Research.

Rosborough G W, Tapley B D (1987) Radial, Transverse and Normal Satellite Position Perturbations due to Geopotential, Celest. Mech. 40, 409.

Scharroo R, Visser P (1998) Precise Orbit Determination and Gravity Field Improvement for the ERS Satellites, J. Geophys. Res. 103, C4, 8113.

Shum CK, Zhang BH, Schutz BE, Tapley BD (1990) Altimeter Crossover Methods for Precison Orbit Determination, J. Astronaut. Sci., 38, 355.

Stammer D, Tokmakian R, Semtner A, Wunsch C (1996) How well does a 1/4 Global Circulation Model Simulate Large-scale Oceanic observations?, J. Geophys. Res. 101, C10, 25,779.

Shum CK, Zhang BH, Schutz BE, Tapley BD (1990) Altimeter Crossover Methods for Precison Orbit Determination, J. Astronaut. Sci., 38, 355.

Tapley BD, and 14 others: (1996) The Joint Gravity Model 3, J. Geophys. Res. 101, B12, 28029.

Wagner CA, Cheney RE (1992) Global Sea Level Change from Satellite Altimetry, J. Geophys. Res. 97, C10, 15607.

Wagner CA, Klokocnik J, Cheney RE, (1997a) Making the Connection between Geosat and TOPEX/Poseidon, J. Geod., 71, 273.

Wagner CA, Klokocnik J, Kostelecky J, (1997b) Dual-Satellite Crossover Latitude-Lumped Coefficients, their use in Geodesy and Oceanography, J. Geod., 71, 603.

Wessel P, Smith WHF (1995) New Version of the Generic Mapping Tools Released, EOS, Trans. AGU, 76, 329.

Figures

Fig. 1. Dual-satellite crossover residuals AA, DD, AD, and DA between Geosat and T/P, JGM  3 - based, in 2x3 deg bins. NOAA 1 year averages. Color scale in centimeters. The residuals "before O.R." (offset removal) and "after O.R." should be distinguished.

Fig. 2. Dual-satellite crossover residuals AA, DD, AD, and DA, all after the coordinate offset removal, between ERS 1 and T/P, JGM  3 - based, in 2x3 deg bins. NOAA 1 year averages. Color scale in centimeters.

Fig. 3. Geosat and T/P: the variable (a,c) and geographically correlated (mean) part (b,d) of the radial orbit error, as projected from the calibrated variance-covariance matrix of the harmonic geopotential coefficients of JGM  3 (to 50x50), using 4 day or 10 day cut of orbit perturbations, with 10x10 deg grid in latitude and longitude. Notice two different scales (both in centimeters).

Fig. 4. ERS 1: the radial orbit error (a,b), its variable (c) and mean (d) parts, as projected from the calibrated variance-covariance matrix of the harmonic geopotential coefficients of JGM 3 (to 50x50), using a 4 day cut of orbit perturbations, with 10x10 deg grid in lat/long. Notice two different scales (centimeters).

Fig. 5. Combinations AA-DD [offset-free by definition] and (AA+DD)/2 [offset-free] of the DSC of Geosat&T/P, JGM  3 - based (a, b), 1 year NOAA averages, with their relevant Student statistics (c, d). The scale (for Figs a, b) is in centimeters, and dimensionless for the statistics (see eq. 23). Clearly visible are systematic effects (the ratio above 1.0) on Fig. 5d (1  year averages) and on Fig. 5e (3  year averages).

Fig. 6. The change in sea level between March and September 1993. Test of Ocean Circulation models: a comparison of seasonal global circulation models with the T/P collinear differences (POCM run  4B and Carton's GCM), in 1x1  deg bin in lat/long. For more details see text.

Fig. 7 Substituting combinations AA+AD and DD+DA of the DSCs of ERS 1&T/P, JGM  3- based (a, b), with their relevant Student statistics (c, d). The scale for Figs a, b is in centimeters, dimensionless for the statistics (see eq. 23). Notice systematic 'along-track' effects (statistics above 1.0) on Figs.  7 c, d. [White spots = no data]. For more details see text.

Fig. 8. Zero (levitating) combinations (AD-AA)/2 and (DD-DA)/2, eqs. 4 and -6, of the DSCs of ERS 1&T/P, JGM  3- based (a, b), 1 year NOAA averages, with their relevant Student statistics (c, d). Notice systematic effects (statistics above 1.0) on Figs. 8c, d, mainly in central Pacific. For more details see text.

Fig. 9. The levitating combinations (DD-DA) and (AD-AA) of ERS  1 and T/P, where the crossovers for ERS  1 are based on DGM  E04 gravity model (Delft) and the crossovers for T/P on JGM  3 (NOAA)! These combinations (Figs.  9 a,b) provide a T/P 'surrogate', according to eq. 17. They are compared to the T/P 'original' single-satellite crossovers (JGM  3 - based), Fig.  9c. A good agreement between Figs.  9a,b and Fig.  9c supports hypothesis that the NOAA data of T/P suffers from an inconsistency probably in media corrections recommended for T/P prior to 1997, in contrast with those used later.

Fig. 10. (a) shows the NOAA (AD-AA) DSC combination, (b) the Pathfinder (AD-AA) DSC combination. These 'surrogates' of T/P are compared to real SSC T/P, (c) from the NOAA GDCs, (d) from Pathfinder altimetry. It again suggests that the NOAA T/P corrections prior to 1997 are inconsistent with those used later.

Contact address:
jklokocn@asu.cas.cz